# Number System for Class 9

To understand the topic Number system for class 9, we need to know all about number system including:

1. What is Number system?
2. What are the types of numbers?
3. Example questions to understand these number types?
4. Practice questions to understand the number system.

## Introduction to number system class 9

The collection of numbers is called Number system

 Natural Numbers N 1, 2, 3, 4, 5, …… Whole Numbers W 0,1, 2, 3, 4, 5…. Integers Z …., -3, -2, -1, 0, 1, 2, 3, … Rational Numbers Q p/q form, where p and q are integers and q is not zero. Irrational Numbers Which can’t be represented as rational numbers

Note: every natural number is an integer and 0 is a whole number which is not a whole number.

### How to find a rational number between two given numbers a and b?

• To find a rational number between a and b, find (a+b)/2.

Example: If we want to find a rational number between 3 and 4, the answer is (3+4)/2 = 7/2.

### Irrational Numbers

A number is called an irrational number if it can’t be represented in a p/q form, where p and q are integers.

### Real Numbers

The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R.

Every real number is a unique point on the number line and also every point on the number line represents a unique real number.

## Difference between Terminating and Recurring Decimals.

 Terminating Decimals Repeating Decimals If the decimal expression of a/b terminates. I.e comes to an end, then the decimal so obtained is called Terminating decimals. A decimal in which a digit or a set of digits repeats repeatedly periodically, is called a repeating decimal. Example: ¼ =0.25 Example: ⅔ = 0.666… = 0.$\bar{6}$

### Some Special Characteristics of Rational Numbers:

• Every Rational number is expressible either as a terminating decimal or as a repeating decimal.
• Every Terminating decimal is a rational number.
• Every repeating decimal is a rational number.

### Irrational Numbers

• The non terminating, non repeating decimals are irrational numbers.

Example: 0.0100100001001…

• Similarly if m is a positive number which is not a perfect square, then $\sqrt{m}$ is irrational.

Example: $\sqrt{3}$

• If m is a positive integer which i snot a perfect cube, then $\sqrt[3]{m}$ is irrational.

Example: $\sqrt[3]{2}$

### Properties of Irrational Numbers:

• These satisfy the commutative, associative and distributive laws for addition and multiplication.
• Sum of two irrationals need not be irrational.

Example: (2 + $\sqrt{3}$) + (4 – $\sqrt{3}$) = 6

• Difference of two irrationals need not be irrational.

Example: (5 + $\sqrt{2}$) – (3 + $\sqrt{2}$) = 2

• Product of two irrationals need not be irrational.

Example: $\sqrt{3}$ x $\sqrt{3}$ = 3

• Quotient of two irrationals need not be irrational.

2$\sqrt{3}$/$\sqrt{3}$ = 2

• Sum of rational and irrational is irrational.
• Difference of rational and irrational is irrational.
• Product of rational and irrational is irrational.
• Quotient of rational and irrational is irrational.

## Real Numbers

A number whose square is non-negative, is called a real number.

• Real numbers follows Closure property, associative law, commutative law, existence of additive identity, existence of additive inverse for Addition.
• Real numbers follows Closure property, associative law, commutative law, existence of multiplicative identity, existence of multiplicative inverse, Distributive laws of multiplication over Addition for Multiplication.

### Rationalisation

If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number, is called rationalisation.

Example:

3/$\sqrt{2}$ = 3/$\sqrt{2}$ x $\sqrt{2}$/ $\sqrt{2}$ = 3$\sqrt{2}$/2

Let a>0 be a real number, and let p and q be rational numbers, then we have:

i) (ap x aq) = a(p+q)

ii) (ap)q = apq

iii)ap /aq = a(p-q)

iv) ap x bp = (ab)p

Example: Simplify (36)½

Solution: (62)½ = 6(2 x ½) = 61 = 6

Class 9 Number system Extra questions:

Q1) The simplest form of 1.$\tilde{6}$ is?

Q2) An irrational number between $\sqrt{2}$ and $\sqrt{3}$ is?

Q3) Give an example of two irrational numbers whose sum as well as the product is rational.

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